Source: Groves et al. 2009

• Measurement
  • Construct validity: the extent to which the measure is related to the underlying construct/concept
  • Measurement error:
    • e.g. underreporting of sensitive behaviors
  • Processing error: i.e. data entry error
• Representation: we want our estimate of certain variable to be as representative (i.e. externally valid) as possible.
  • Coverage error: the nonobservational gap between the target population and the sampling frame
    • e.g. sampling through phone calls can make the resulting sample less representative.
  • Sampling error: non observational gap between the sampling frame and the sample
    • e.g. some members of the sampling frame are given smaller chance of selection
  • Non-response error
    • Unit non-response
    • Item non-response
  • Adjustment error

\begin{aligned} \text{Corr}(x, y) =& \frac{\text{Cov}(x, y)}{\sigma_x \sigma_y} \\ =& \frac{\frac{1}{n-1}\sum_{i=1}^n [(x_i - \bar{x}) (y_i - \bar{y})]}{\sigma_x \sigma_y} \\ =& \frac{1}{n-1}\sum_{i=1}^n [\text{z-score}(x) \times \text{z-score}(y)] \end{aligned}

• Covariance: To what extent the two variables covary on average.
• Correlation: covariance of the two variables rescaled by their standard deviations.

Overview

• Research Question: Does having daughters cause judges to vote in a pro-feminist direction?
  • Study type: observational studies
  • Explanatory variable: whether a judge has at least one daughter.
  • Outcome variable: progressive.vote - The proportion of the judge’s votes on women’s issues which were decided in a pro-feminist direction.
• Potential Confounders
  • a judge’s gender (Q2)
  • a judge’s party identification (Q2 & Q3)
  • whether a judge has at least one child (Q3)
  • the number of children (Q4)
  • other confounders? Can we assess them in the existing data? (Q5)
• The goal of each question
  • Q1: summary statistics and check balance
  • Q2: association between progressive.vote and two confounders - republican and woman
  • Q3: association between progressive.vote and two confounders - whether a judge has at least one child and republican
  • Q4: association between progressive.vote and explanatory variable - whether a judge has at least one daughter, conditional on the total number of children
    • Considering confounding bias and covariate balance, should we include judges who have no child?
  • Q5: assess the plausibility of unconfoundedness assumption
    • Is the number of children correlated with both children’s gender and progressive.vote?
    • Is any other pre-treatment variables potentially correlated with both children’s gender and progressive.vote?

Question 1

• The total number of judges; gender composition and party composition
• Party composition between male and female judges (Hint: create a contingency table showing proportion)
• Range of the outcome variable progressive.vote.
• Interpret the result.

Question 2

• Create a boxplot with one single command to compare the distribution of progressive.vote across 4 groups: Republican men, Republican women, Democratic men, Democratic women. (Hint: y ~ x1 + x2 in boxplot())
• Interpret the result.

Question 3

• Create a binary variable that denotes if a judge has at least one child.
• Compare the distribution of this binary variable between Republican and Democratic judges. (Hint: difference in means)
• Compute the difference in means of progressive.vote between judges who have at least one child and those who don’t.
• Compute the difference in means of progressive.vote between Republican and Democratic parents.
  • Hint: compute difference in means for each group, or use tapply(..., list(..., ...), mean)
  • e.g. Republican parents: judges who are republican and who have at least one child

Question 4

• Compute difference in means of progressive.vote between judges who have at least one daughter and those who don’t have any.
• Repeat the same computation across judges with different number of children.
  • only focus on judges with 1, 2, and 3 children.
  • Hint: compute difference in means for each group, or use tapply(..., list(..., ...), mean)
• What assumption do we need to interpret previous results?

Question 5

Conditional on the number of children, the number of daughters a judge has is random. How can we evaluate the validity of this assumption?

• Compare the means of girls across judges with different number of children.
• Compare the means of girls across judges, divided by other potential confounders in the data.